Puyat, Pagod, Physics: eight

For a better understanding of how I did this activity, refer to the Supplementary Information Section where the important parts of the codes I used are shown.

In the world of image processing, the Fourier Transform is the star quarterback. Well, I’m not really sure about that but the Fourier Transform pops up a lot. This activity studies the enhancement within the frequency domain through (you guessed it) the Fourier Transform.

First off, we look into the Convolution Theorem. The Fourier Transform of an image comprising of two dots located along the x-axis was taken. The said Fourier Transform resembles a sinusoid of sorts.

Figure 1. (A) Two dots and (B) their Fourier Transform.

Now, we replace those two dots with circles, squares and Gaussians whose radii, width and variance values are varied.

Figure 2. Circles of radii (A) 0.01, (B) 0.05, (C) 0.10, (D) 0.15, (E) 0.20 and (F) 0.25

Figure 3. Fourier Transforms of circles of radii (A) 0.01, (B) 0.05, (C) 0.10, (D) 0.15, (E) 0.20 and (F) 0.25

Figure 4. Squares of width (A) 0.01, (B) 0.05, (C) 0.10, (D) 0.15, (E) 0.20 and (F) 0.25

Figure 5. Fourier Transforms of squares of width (A) 0.01, (B) 0.05, (C) 0.10, (D) 0.15, (E) 0.20 and (F) 0.25

Figure 6. Gaussians with variance of (A) 0.01, (B) 0.05, (C) 0.10, (D) 0.15, (E) 0.20 and (F) 0.25

Figure 7. Fourier Transforms of Gaussians with variance of (A) 0.01, (B) 0.05, (C) 0.10, (D) 0.15, (E) 0.20 and (F) 0.25

As you can see, when the radii, width and variance values are increased, the area spanned by their Fourier Transforms shrinks. Mathematically, an area of let’s say A is mapped as 1/A in the Fourier space. So in a way, there was no need for the panic and confusion I went through.

When convolving a pattern onto ten randomly distributed points, what I found was that the points’ positions remain the same but the points themselves take on the appearance of the pattern.

Figure 8. (A) Ten randomly distributed dots, (B) a pattern and (C) the result of their convolution.

Lastly (for this section), we investigate the effect of distances between dots on the Fourier Transform of images. What I observed here was similar to the phenomena with the increased radii, etc. The greater separation distance (and/or density) of dots in the image, the space that is occupied by the Fourier Transform increases too.

Figure 9. Grids with spacing of (A) 10, (B) 25, (C) 50, (D) 75, (E) 100 and (F) 125.

Figure 10. Fourier transforms of grids with spacing of (A) 10, (B) 25, (C) 50, (D) 75, (E) 100 and (F) 125.

We now go into Ridge Enhancement and whaddayaknow, we study something I’ve been all too familiar with since my grade school days when I started to watch CSI. I’ve always known that the technology they show in these criminal periodicals isn’t really at par with the technology that us mere mortals have to wrestle with. But in the next part, I experience this face-to-face (and I seriously wish I hadn’t).

Figure 11. (A) Gray scale image of an unbinarized fingerprint image and (B) its Fourier Transform.

Figure 12. (A) The mask used and the (B) masked Fourier Transform.

Figure 13. (A) The filtered fingerprint image and (B) its binarized version.

This is definitely not CSI material. I mean, you wouldn’t have narrowed your suspect pool by a lot using this print. Out of curiosity, I even realized that this could be my left thumb print which, if it were really the case, I just caught myself. (What?)

For the Line Removal section, we use an image that I should have gotten from here but their website was uncooperative. After a slight complaint on Twitter though, BA e-mailed me the picture. Nevertheless, the result turned out pretty well if I do say so myself..

Figure 14. (A) Gray scale image and (B) its corresponding Fourier Transform.

Figure 15. (A) The mask used, (B) masked Fourier Transform and (C) the resulting image.

Lastly, the Canvas Weave Modeling and Removal section tasked us to remove the canvass patterns in an image which – honestly, I tried everything – I was not able to do. What follows are the best results I got. I think the crappiness of my results has much to do with my mask. I originally saved it as .PNG then converted it to .BMP and life hasn’t been the same since then. Sigh.